Most hunters take ethics in hunting very seriously
from before the shot to after,
part of such is practice, using the right firearm for the right type of hunting you plan on doing,

first question is why do you want to hunt?
people hunt for different reasons,
food,
population control,
animal management,
critter control,
air rifle is ok for small game, squirrel, rabbits,
even up to coyote with a well placed shot,
but dont recomend it,
air rifles have come a long way, but are not as accurate off the shelf as a good .22 rifle or .28 ga shotgun
you will injure more game with an air rifle than you would with a dedicated firearm.
you will not eliminate the possibility of injuring animals,
all you can do is minimize the chance of that happening through practice, appropriate firearm, and choice of shots.
in time chances are you will injure an animal,
you will feel bad,
but have to know yourself and know you took precautions to avoid that, and learn from your mistake to try to avoid it from happening again.